I would like to know if anticommuting fields (which physicists use as fermions) emerge naturally from the spin representation theory of $SO(d-1,1)$. Is the fact that spinor fields anticommute a physics result or a math result?
We can build non-spin representations of $SO(d-1,1)$ out of the fundamental vector representation. The vector space $V$ on which this representation acts is just Mink$_d$. We can define scalar, vector, tensor, etc. fields on $V$ as usual, $$\phi(x),A_{\nu}(x), T_{\mu\nu}(x), \text{etc}.\qquad x\in V.$$
These fields commute $\phi(x) A_{\nu}(x) = A_{\nu}(x) \phi(x)$.
We built spin representations of $SO(d-1,1)$ (following Harris & Fulton) by first constructing the Clifford algebra
$$C(d-1,1) = T^*(V)/ I(d-1,1)$$ where $$T^*(V)=\bigoplus_{n \geq 0}V^{\otimes n} $$ and $I(d-1,1)$ is the ideal generated by elements of the form $\{v^{\mu},w^{\nu}\} - 2\eta^{\mu\nu}$ where $v,w \in V$. For instance if $\eta^{\mu\nu} = 0$ then $C(d-1,1)$ is just the exterior algebra; but we take $\eta^{\mu\nu}$ to have signature $(d-1,1)$. Then we can define the usual Dirac gamma matrices $\Gamma^{\mu}$ and $[\Gamma^{\mu},\Gamma^{\nu}]$ furnishes a representation of $SO(d-1,1)$.
Now, as for non-spin, I presume we can define spinor fields $\eta(c)$ where $c$ belongs to the $2^{d/2}$ dimensional space $C(d-1,1)$. Is there some consistency condition which implies that two such fields satisfy
$$\eta(c) \chi(c') = -\chi(c') \eta(c)$$
directly from the definition of $C(d-1,1)$ perhaps? Why must $\eta,\chi$ be anticommuting in stark contrast with $\phi, A_{\nu}, T_{\mu\nu}$?
